Theorem elcncf1di 12774

Description: Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)

Hypotheses

Ref	Expression
elcncf1d.1	|- ( ph -> F : A --> B )
elcncf1d.2	|- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
elcncf1d.3	|- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )

Assertion

Ref	Expression
elcncf1di	|- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Distinct variable groups:   x, w, y, A   w, B, x, y   w, F, x, y   ph, w, x, y   w, Z

Allowed substitution hints:   Z( x, y)

Proof of Theorem elcncf1di

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcncf1d.1	. . 3 |- ( ph -> F : A --> B )
2		elcncf1d.2	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
3	2	imp 123	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> Z e. RR+ )
4		an32 552	. . . . . . . . 9 |- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) <-> ( ( x e. A /\ y e. RR+ ) /\ w e. A ) )
5	4	anbi2i 453	. . . . . . . 8 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
6		anass 399	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
7	5, 6	bitr4i 186	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) )
8		elcncf1d.3	. . . . . . . 8 |- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
9	8	imp 123	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
10	7, 9	sylbir 134	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
11	10	ralrimiva 2508	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
12		breq2 3941	. . . . . 6 |- ( z = Z -> ( ( abs ` ( x - w ) ) < z <-> ( abs ` ( x - w ) ) < Z ) )
13	12	rspceaimv 2801	. . . . 5 |- ( ( Z e. RR+ /\ A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
14	3, 11, 13	syl2anc 409	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
15	14	ralrimivva 2517	. . 3 |- ( ph -> A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
16	1, 15	jca 304	. 2 |- ( ph -> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
17		elcncf 12768	. 2 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) ) )
18	16, 17	syl5ibrcom 156	1 |- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1481   A.wral 2417   E.wrex 2418   C_ wss 3076   class class class wbr 3937   -->wf 5127   ` cfv 5131  (class class class)co 5782   CCcc 7642   < clt 7824   - cmin 7957   RR+crp 9470   abscabs 10801   -cn->ccncf 12765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-map 6552  df-cncf 12766

This theorem is referenced by:  elcncf1ii  12775  cncfmptc  12790  cncfmptid  12791  addccncf  12794  negcncf  12796